nLab
Tomasz Maszczyk

Tomasz Maszczyk is a Polish mathematician (Warsaw University and IMPAN) with main interests in algebraic geometry, noncommutative algebra and noncommutative geometry (both of algebraic and of operator-algebraic flavour). With Piotr Hajac, he coordinates a weakly active seminar in Warsaw on noncommutative geometry at IMPAN, frequently featuring international leading experts in noncommutative geometry.

Maszczyk created an original viewpoint and approach to noncommutative geometry based on monoidalabelian categories viewed as categories of quasicoherent sheaves. Galois and reconstruction theorems of various sort (Tannakian, Morita-type…), and remarkable and consistent application of categorical thinking (neglected by most of the mainstream schools in noncommutative geometry), play role at many places in his work. His works gave light to a number of problems related to cyclic homology, corings, Hopf-Galois extensions, descent theory and mathematics of regular differential operators in commutative and noncommutative algebraic geometry.

Most of Maszczyk’s main program is still unpublished (even on arXiv). Among his articles on the arXiv cf.

According to Grothendieck-Galois theory, there is a close relation between splittings of commutative rings by an appropriate base change and (groupoid) actions. The reconstruction of the action from a given splitting is called the Galois reconstruction. According to Grothendieck-Deligne-Saavedra Rivano-Tannaka theory, there is another close relation between representations of a given groupoid and the groupoid itself. The reconstruction of the groupoid from its representations is called the Tannaka reconstruction. We show that both reconstructions are particular cases of our theorem about splittings of flat covers in the bicategory of monoidal categories.